Abstract
Let \(f: X\to \mathbb {I}\) be a map with A = f −1(0) ≠ ∅ and fix a point a∉X. Endow \(\widehat {X}= (X\setminus A)\cup \{a\}\) with the topology generated by the open sets of X ∖ A together with all sets of the form \(\widehat {X}\setminus f^{-1}[\frac {1}{n}, 1]\), where \(n\in \mathbb {N}\). Clearly, X ∖ A and \(\widehat {X} \setminus \{a\}\) are homeomorphic spaces and it is readily seen that if X is T 1 or regular, then so is \(\widehat {X}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
E.K. van Douwen, The small inductive dimension can be raised by the adjunction of a single point. Indag. Math. 35, 434–442 (1973)
T.C. Przymusiński, A note on the dimension theory of metric spaces. Fund. Math. 85, 277–284 (1974)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Charalambous, M.G. (2019). No Finite Sum Theorem for the Small Inductive Dimension of Metrizable Spaces. In: Dimension Theory. Atlantis Studies in Mathematics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-22232-1_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-22232-1_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22231-4
Online ISBN: 978-3-030-22232-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)